MATH 161

FINAL EXAM

 

  1. (10 pt) a) Find the area between the curves and .

    2. b) Find the volume of obtained by revolving the region bounded by these two curves about the line .

     

  2. (5 pt) Let be a continuous odd function (). Find the average value of on the interval [-a, a].
  3. (15 pt) Evaluate the following integrals:

    4.  

    a) b) c)

     

  4. (10 pt) a) Use Simpson’s Rule with n=4 to estimate .

    5. b) Estimate the error for using the midpoint rule and Simpson’s Rule with n=4 to approximate .

     

  5. (5 pt) Find the arclength along from (0,0) to (4,8).

    6.  

  6. (10 pt) Solve the differential equation:

    7.  

     

  7. (10 pt) Consider the following parametric equation:

  1. Sketch this ellipse and find its area.
  2. Write an integral that expresses the surface area obtained by spinning the upper half of this ellipse about the x-axis. After you have done this, let
and solve the integral.

 

  1. (10 pt) Sketch the graph of the polar equation and find its length from 0 to p . (Hint: )
  2. (15 pt) Determine of the following series converge or diverge.

    3. a) b) c)

     

  3. (10 pt) a) Find the Taylor series about of
. What is the interval of convergence?

b) Use the Maclaurin series for to find with error less than .001.

 

 

USEFUL FORMULAS:

 

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